Appendix B: Parameter Estimation - Revision history

Lisa Hacker at 19:19, 15 September 2023

2023-09-15T19:19:24Z

← Older revision		Revision as of 19:19, 15 September 2023
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	{{template:ALTABOOK ~~SUB~~\|Appendix B\|Parameter Estimation}}		{{template:ALTABOOK\|Appendix B\|Parameter Estimation}}

	This appendix presents two methods for estimating the parameters of accelerated life test data analysis models (ALTA models). The graphical method, which is based on probability plotting or least squares (Rank Regression on X or Rank Regression on Y), has some limitations. Therefore, the Maximum Likelihood Estimation (MLE) method is used for all parameter estimation in ALTA.		This appendix presents two methods for estimating the parameters of accelerated life test data analysis models (ALTA models). The graphical method, which is based on probability plotting or least squares (Rank Regression on X or Rank Regression on Y), has some limitations. Therefore, the Maximum Likelihood Estimation (MLE) method is used for all parameter estimation in ALTA.

Sharon Honecker: /* Example of Graphical Method for Accelerated Life Data */

2017-02-06T16:53:15Z

Example of Graphical Method for Accelerated Life Data

← Older revision		Revision as of 16:53, 6 February 2017
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	& [{{\widehat{\beta }}_{1}}= & 3.8,\text{ }{{\widehat{\eta }}_{1}}=6692] \\		& [{{\widehat{\beta }}_{1}}= & 3.8,\text{ }{{\widehat{\eta }}_{1}}=6692] \\
	& \text{ }\!\![\!\!\text{ }{{\widehat{\beta }}_{2}}= & 4.2,\text{ }{{\widehat{\eta }}_{2}}=5716] \\		& \text{ }\!\![\!\!\text{ }{{\widehat{\beta }}_{2}}= & 4.2,\text{ }{{\widehat{\eta }}_{2}}=5716] \\
	& [{{\widehat{\beta }}_{3}}= & 4,\text{ }{{\widehat{\eta }}_{3}}=4774]		& [{{\widehat{\beta }}_{3}}= & 4.0,\text{ }{{\widehat{\eta }}_{3}}=4774]
	\end{align}\,\!</math>		\end{align}\,\!</math>

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	where <math>L=\eta \,\!</math>, (- <math>\ln K)\,\!</math> is the intercept, and (- <math>n)\,\!</math> is the slope of the line.		where <math>L=\eta \,\!</math>, (<math>-\ln K)\,\!</math> is the intercept, and (<math>-n)\,\!</math> is the slope of the line.

	The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:		The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:
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	::<math>\begin{align}		::<math>\begin{align}
	& \widehat{K}=\ \frac{1}{6000\~~text{ }~~{{403}^{4.5272}}} =\ 2.67\cdot {{10}^{-16}}		& \widehat{K}=\ \frac{1}{6000\cdot{{403}^{4.5272}}} =\ 2.67\cdot {{10}^{-16}}
	\end{align}\,\!</math>		\end{align}\,\!</math>

Nicolette Young: /* Example of Graphical Method for Accelerated Life Data */

2015-12-21T17:49:26Z

Example of Graphical Method for Accelerated Life Data

← Older revision		Revision as of 17:49, 21 December 2015
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	*<math>{{\widehat{\beta }}_{3}},\,\!</math> <math>{{\widehat{\eta }}_{3}}\,\!</math> are the parameters of the 423 psi data.		*<math>{{\widehat{\beta }}_{3}},\,\!</math> <math>{{\widehat{\eta }}_{3}}\,\!</math> are the parameters of the 423 psi data.

	[[Image:ALTAProbabilityplot.png\|center\|~~500px~~\|]]		[[Image:ALTAProbabilityplot.png\|center\|600px\|]]

	Since the shape parameter, <math>\beta ,\,\!</math> is not common for the three stress levels, the average value is estimated.		Since the shape parameter, <math>\beta ,\,\!</math> is not common for the three stress levels, the average value is estimated.
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:ALTAProbabilityplot2.png\|center\|~~700px~~\|]]		[[Image:ALTAProbabilityplot2.png\|center\|600px\|]]

	The IPL relationship is given by:		The IPL relationship is given by:

Nicolette Young: /* Example of Graphical Method for Accelerated Life Data */

2015-12-21T17:49:04Z

Example of Graphical Method for Accelerated Life Data

← Older revision		Revision as of 17:49, 21 December 2015
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:ALTAProbabilityplot2.png\|center\|~~600px~~\|]]		[[Image:ALTAProbabilityplot2.png\|center\|700px\|]]

	The IPL relationship is given by:		The IPL relationship is given by:

Nicolette Young: /* Graphical Method */

2015-12-21T17:26:54Z

Graphical Method

← Older revision		Revision as of 17:26, 21 December 2015
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	The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.		The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.

	[[Image:ALTAB.1.png\|center\|~~500px~~]]		[[Image:ALTAB.1.png\|center\|600px]]
	[[Image:ALTAB.1.1.png\|center\|~~500px~~]]		[[Image:ALTAB.1.1.png\|center\|600px]]

	==Example of Graphical Method for Accelerated Life Data==		==Example of Graphical Method for Accelerated Life Data==
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:ALTAProbabilityplot2.png\|center\|~~500px~~\|]]		[[Image:ALTAProbabilityplot2.png\|center\|600px\|]]

	The IPL relationship is given by:		The IPL relationship is given by:
Line 90:		Line 90:
	The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:		The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:

	[[Image:ALTAlifevsstress.png\|center\|~~500px~~\|]]		[[Image:ALTAlifevsstress.png\|center\|600px\|]]

	The slope of the line is the <math>n\,\!</math> parameter, which is obtained from the plot:		The slope of the line is the <math>n\,\!</math> parameter, which is obtained from the plot:

Lisa Hacker: /* Example of Graphical Method for Accelerated Life Data */

2014-02-10T23:21:34Z

Example of Graphical Method for Accelerated Life Data

← Older revision		Revision as of 23:21, 10 February 2014
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	*<math>{{\widehat{\beta }}_{3}},\,\!</math> <math>{{\widehat{\eta }}_{3}}\,\!</math> are the parameters of the 423 psi data.		*<math>{{\widehat{\beta }}_{3}},\,\!</math> <math>{{\widehat{\eta }}_{3}}\,\!</math> are the parameters of the 423 psi data.

	[[Image:ALTAProbabilityplot.png\|center\|~~400px~~\|]]		[[Image:ALTAProbabilityplot.png\|center\|500px\|]]

	Since the shape parameter, <math>\beta ,\,\!</math> is not common for the three stress levels, the average value is estimated.		Since the shape parameter, <math>\beta ,\,\!</math> is not common for the three stress levels, the average value is estimated.

Lisa Hacker: /* Example of Graphical Method for Accelerated Life Data */

2014-02-10T23:21:19Z

Example of Graphical Method for Accelerated Life Data

← Older revision		Revision as of 23:21, 10 February 2014
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:ALTAProbabilityplot2.png\|center\|~~400px~~\|]]		[[Image:ALTAProbabilityplot2.png\|center\|500px\|]]

	The IPL relationship is given by:		The IPL relationship is given by:
Line 90:		Line 90:
	The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:		The values of eta obtained previously are now plotted on a log-linear scale yielding the following plot:

	[[Image:ALTAlifevsstress.png\|center\|~~400px~~\|]]		[[Image:ALTAlifevsstress.png\|center\|500px\|]]

	The slope of the line is the <math>n\,\!</math> parameter, which is obtained from the plot:		The slope of the line is the <math>n\,\!</math> parameter, which is obtained from the plot:

Lisa Hacker: /* Graphical Method */

2014-02-10T23:20:47Z

Graphical Method

← Older revision		Revision as of 23:20, 10 February 2014
Line 8:		Line 8:
	The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.		The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.

	[[Image:ALTAB.1.png\|center\|~~400px~~]]		[[Image:ALTAB.1.png\|center\|500px]]
	[[Image:ALTAB.1.1.png\|center\|~~400px~~]]		[[Image:ALTAB.1.1.png\|center\|500px]]

	==Example of Graphical Method for Accelerated Life Data==		==Example of Graphical Method for Accelerated Life Data==

Nicolette Young: /* Graphical Method */

2014-02-10T22:23:15Z

Graphical Method

← Older revision		Revision as of 22:23, 10 February 2014
Line 8:		Line 8:
	The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.		The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (e.g., Arrhenius, inverse power law, etc.). To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as BX% life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line.

	[[Image:ALTAB.1.png\|center\|~~650px~~\|]]		[[Image:ALTAB.1.png\|center\|400px]]
			[[Image:ALTAB.1.1.png\|center\|400px]]

	==Example of Graphical Method for Accelerated Life Data==		==Example of Graphical Method for Accelerated Life Data==

Richard House at 22:41, 21 September 2012

2012-09-21T22:41:22Z

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